A parity map of framed chord diagrams

作者：Ilyutko Denis Petrovich^*; Manturov Vassily Olegovich

来源：Journal of Knot Theory and Its Ramifications, 2015, 24(13): 1541006.

DOI：10.1142/S0218216515410060

• Chord diagram
• framing
• weight system
• module

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module M-2 which is generated by chord diagrams on two circles and factored by 4T-relations. Then we construct a "parity" map from the module of framed chord diagrams into M-2 and a weight system on M-2. Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

• 出版日期2015-11